MECH 422
Do all 3 problems
Problem 1 (33 points): What is the heat flux out of the right face of this element? You
may assume that the conductivity k = 1. Use rigorous finite element formulas, not estimates.
T=0
T=0
T=0
8
T=0
T=6
T=0 T=0
T=0
T=0
4
Problem 2 (33 points): Consider the isoparametric finite element with conductivity k = 1
shown below:
y
(0,3)
(1,3)
2
1
3
4
(0,0)
(1,0)
X
Set up the integral for K13 . (The entry of the stiffness matrix in the first row, third column.
Set up the integral as completely as possible, but do not perform any integrations.)
1
Problem 3 (34 points): Consider radial heat flow in a cylinder of unit depth (into the
page), for which the weak form is
Z
Z
dT
dT
k 2πx dx = T 2πxg dx
dx
dx
(a) What is the stiffness matrix for a general two-node finite element for this case? Set up
all integrals as far as possible, but do not integrate.
(b) What is the forcing vector due to internal heat generation for a general two-node finite
element for this case? Set up all integrals as far as possible, but do not integrate.
2